All Questions
Tagged with pr.probability measure-theory
823 questions
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Measurability of a map involving probability measures
Let $X$ be a metrizable topological space and $\mathscr B_X$ the Borel $\sigma$-algebra on it. Let $\Delta X$ denote the set of probability measures on $(X,\mathscr B_X)$, and let $\mathscr B_{\Delta ...
- 405
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66
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Does convergence in probability of iid samples imply convergence in measure of the sampled functions?
Let $g_i: [0, 1] \to \mathbb R$ be $L^1$ functions, equibounded in $L^1$ norm. Let $X_i$ a sequence of iid uniform random variables on $[0, 1]$. Suppose that
$$\frac{1}{n} \sum_{i = 1}^n g_i (X_i) \to ...
- 6,165
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500
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Uncountable families of measurable sets with pairwise positive intersections
Let $(X,\mathcal{B},\mu)$ be an arbitrary finitely additive probability measure space, let $a>0$ and let $(A_i)_{i\in I}$ be an uncountable family of subsets with measure $\geq a$.
Is there an ...
- 10.6k
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Non-atomic probability measures on N
One can intuitively imagine picking a random natural number and ask to what extent the intuition can be axiomatized.
Using the axiom of choice, there is a total finitely additive (monotonic) averaging ...
- 7,545
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129
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A Talagrand inequality for the supremum of partial sums over function classes under dependence. (Reference request)
As a consequence to the Talagrand concentration inequality, it is well known that for a measurable space $(S,\mathcal{S})$ and an i.i.d. sample $X_1,...,X_n$ of $S$-valued random variables, if $\...
- 141
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188
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Girsanov's theorem for Gaussian measures as the Cameron-martin theorem with a random shift
Let $H \subset E$ be the Cameron-Martin space of a Gaussian measure $\mu$ on a separable Banach space $E$. The Cameron-Martin theorem states that for all $h \in E$ we have $h \in H$ if and only if $\...
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183
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What is a natural interpretation of the commutator of the conditional expectation operator?
Notation: We denote by $\mathbb E_{\mathcal F} X$ the conditional expectation of the random variable $X$ with respect to the $\sigma$-algebra $\mathcal F$.
Given two $\sigma$-algebras $\mathcal G, \...
- 6,165
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61
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Bound on $\int_0^1\sqrt{\log N_{[]}(\varepsilon,\mathcal{F},d)} \, d\varepsilon$ over the class of half-spaces $\mathcal{F}$ on $\mathbb{R}^d$?
For a class of functions $\mathcal{F}$ and a pair $f,g\in\mathcal{F}$ with $f\leq g$, the interval $[f,g]=\{h:f(x)\leq h(x)\leq g(x),\forall x\in\mathbb{R}^d\}$ is called a bracket for $\mathcal{F}$. ...
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92
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Existence of ergodic subgroup invariant to a product measure
Let $X=\{0, 1\}^{\mathbb{N}}$ and $G$ be the group of permutations, each of which only permutes finitely many coordinates of $X$. Fix a sequence $(\lambda_n)_{n\in \mathbb{N}} \subseteq (0, 1]$ and ...
- 307
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215
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Compactness with respect to topology induced by total-variation distance
I've been working on a problem and at some point in the proof I need to show that the following set $$\left\{\mu \in \mathcal{P}_{ac}(\mathbb R^d): \int \varphi(x)\mu(\mathrm{d}x)\leq C\right\}$$
is ...
- 21
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145
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What is an example of a non-tight probability measure?
Billingsley (Convergence of Probability Measures, 1968) and van der Vaart and Wellner (Weak Convergence and Empirical Processes, 2023) discuss the concept of tight probability measures and use the ...
- 277
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45
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Small deviation asymptotics for sub-gaussian diffusions in dirichlet spaces
Let $(X,d,\mu)$ be a metric measure space equipped with a strongly local, regular Dirichlet form $(\mathcal{E}, \mathcal{D}(\mathcal{E}))$ on $L^2(X,\mu)$. Assume that the associated heat kernel $p_t(...
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185
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Sum of $X_k$ with $\mathbb{P}(X_k=\pm 1) = 1/2\pm 1/(2\sqrt{k})$
Let $\{X_k\}$ be a sequence of mutually independent random variables with
\begin{align}
\mathbb{P}(X_k = 1) & = \frac{1}{2} + \frac{1}{2\sqrt{k}},
\\
\mathbb{P}(X_k = -1) & = \frac{1}{2} - \...
- 269
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205
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When should the empirical measure of an infinite sequence be defined?
Let $(x_n)_{n \in \mathbb{N}}$ be a (deterministic) sequence of nonnegative reals, possibly even with $x_n \in \mathbb{N}$ if you prefer. Then we'd like to define the empirical measure of such a ...
- 6,406
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176
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Are measurable maps with countably separated image in a Banach space always strongly measurable?
Let $(E,\|.\|)$ be a (not necessarily separable) Banach space and $\Sigma_E$ the Borel $\sigma$-algebra (w.r.t. the norm topology). Let $(\Omega,\Sigma_\Omega)$ be a measurable space (which we can ...
- 285
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991
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Choosing a relative large density subsequence from a low density sequence
My question is somewhere in the interface of combinatorics, probability, and measure theory. It is quite ad-hoc, and I wonder if there is a counter example.
Consider for example the unit interval $[0,...
- 502
4
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198
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When a null uncountable set can be image of some increasing function with discontinuities on a dense countable set
Consider the following result:
A: Let $f:D \to \mathbb R$ be an increasing function with discontinuities on a dense countable subset of $D$ such that the jump values sum to $\mu(D)$, where $D$ is a ...
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370
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Convergence of iterated conditional expectations
Notation: We write $\mathbb E_{\mathcal F} X$ for the conditional expectation $\mathbb E[X|\mathcal F]$ of a random variable $X$ with respect to a $\sigma$-algebra $\mathcal F$.
Let $X$ be an ...
- 6,165
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29
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Steiner symmetrization of smooth function on non-simply connected regions
Given a smooth function $u$ defined on $\mathbb{R}^2$, restrict $u$ to a subset $\Omega \subset \mathbb{R}^2$ (possibly not simply connected) foliated by level sets of a smooth function $\psi: \Omega \...
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Convergence in $\mathbb{L}_1$ implies convergence "perturbed" conditional expectations
Consider a sequence of conditional pdf's $p_n(y | x)$ on a Polish space $X \times Y$, endowed with its Borel sigma algebra. Suppose, as $n\rightarrow \infty$, in $\mathbb{L}_1$ (the following ...
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-1
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2
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250
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$p$-norm of random variables and weighted $L^p$ space resemblance
I noticed a very similar relationship between weighted $L^p$ space (denoted $L_w^p$) and normed vector space of random variables. I want to unify these two spaces but there always seems to be a ...
- 1
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3
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560
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Why do we need to define a random variable as a function?
I recently learned the mathematical definition of a random variable, namely:
A random variable is a measurable function $X: \Omega \rightarrow \mathbb{R}$ whose domain $\Omega$ is equipped with a $\...
- 27
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1
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170
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Law of large numbers for a continuum of Bernoullis
Suppose I have a family of $n$ independent Bernoulli random variables described by a vector of parameters $(p_i)_{i=1}^n$. As it is well known, the number of successes within this family is a random ...
2
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1
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170
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Lower bound on the Lévy-Prokhorov metric for normal distributions
Let $\mathfrak M(\mathbb R^n)$ denote the metric space of probability measures (over $\mathbb R^n)$ equipped with the Lévy-Prokhorov metric $\rho$. Consider two $n$-variate normal distributions $\...
- 103
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45
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Modifiying a sequence of measures to assign a certain value when integrating a fixed function?
Let $f:\mathbb R ^d\to \mathbb R$ be some continuous function, $|f|\leq A(1+|x|)$, where $|\cdot|$ denotes the usual Euclidean norm. Fix a measure $\mu$ and constant $C$.
Assume that $\mu_n$ is a ...
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58
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Gaussian Hypercontractivity of Chaos based on Gaussian with value in Hilbert spaces?
The classical Gaussian hypercontractivity is stated as following: Suppose $\xi$ is a Gaussian variable and $H_n(\xi)$ is the space of n-th homogeneous Wiener chaos constructed from $\xi$, then for any ...
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114
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Is a $\sigma$-algebra generated by complete independent $\sigma$-algebras also complete?
$
\newcommand{\cA}{\mathcal{A}}
\newcommand{\cB}{\mathcal{B}}
\newcommand{\sP}{\mathscr{P}}
$
Let $(\Omega, \cA, \mu)$ be a probability space and $\cA_1, \cA_2$ sub $\sigma$-algebras of $\cA$. Let $\...
- 835
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79
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Continuity of disintegrations in non locally compact spaces
Let $X$ and $Y$ be Radon spaces, $\mu$ a Borel probability measure on $X$, $F\colon X\to Y$ measurable. Then the disintegration theorem gives us a disintegration $\{\mu^y\}_{y\in Y}$ of $\mu$ with ...
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How probability-rich is the $\sigma$-algebra generated by a sequence of sets? (Sierpiński's theorem on non-atomic measures without using the AoC.)
$\newcommand\F{\mathcal F}\newcommand\si{\sigma}\newcommand\Om{\Omega}\newcommand\ep{\varepsilon}$Let $p\in(0,1)$ and let $(\Om,\F,P)$ be a probability space. Let $(A_n)$ be a sequence in $\F$ such ...
- 128k
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Extracting a subsequence Cesàro converging to the limsup of the Cesàro sums
Let $X_n$ be a sequence of uniformly bounded random variables — that is, there exists some $K > 0$ such that $|X_n| \leq K$ almost surely for all $n \in \mathbb N$.
Write $\bar X_N := \frac{1}{N} \...
- 6,165
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80
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Measurable Extension
Let $(\Omega, \mathcal{F})$ be a measurable space and $X$ some metric space (probably Polish) with the Borel $\sigma$-algebra and a function $f: \Omega \times X \to \mathbb{R}$. Usually, functions ...
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100
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Projection on a countable union of linear subspace
For any natural number $n$, $V_n$ denotes a closed linear subspace of a $L_2(m)$ space, which is an Hilbert Space, where $m$ denotes a finite measure. Moreover $(V_n)$ is increasing, that is $V_n$ is ...
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112
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Smallest ensemble of sets stable by any intersections and finite union
Let $E$ be a set, and let $S$ be any set of subsets of $E$, such that $S$ contains the empty set. Can you identify the smallest set of subsets $T$ of $E$ such that $T$ contains all elements of $S$, ...
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250
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Existence of a positive measurable set with disjoint preimage under iterated transformation
Let $(X,\mathcal B,\mu)$ be a atomless probability measure space and $T:X\to X$ be a non-singular transformation such that $\mu\left({x\in X: T^n(x)=x}\right)=0$ for every $n\ge 1$. Let $A\in \mathcal ...
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Reference on multifractal complex measures?
This is a cross-post of this physicsSE post; I am also posting it here since this question lies at the boundary of both physics and math.
I am learning about multifractal formalism recently. It seems ...
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Prokhorov's theorem for countably many random measures on a Polish space
I am looking for help to show the following lemma:
Lemma Let $(\Omega,\mathcal A,\mathbb P)$ be a complete, standard Borel probability space and $\mathcal X$ a Polish space. Let $\mathcal P(\mathcal ...
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271
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Expectation on a Polish space
I was wondering, if given a Polish space $X$, and given some probability measure $p$ on $X$, can the expectation of an $X$-valued function be taken? In particular, would the integral
$\int_X x dp$ ...
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Sufficient condition for the graph of a measurable map to be measurable
Let $f:X \to Y$ be measurable map between measurable spaces w.r.t. to their corresponding $\sigma$-algebras $\Sigma_X$ and $\Sigma_Y$, resp.
If $(X,\Sigma_X)$ is a standard Borel space can we always ...
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73
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Criteria giving sufficient conditions for a Borel measure to have compact support
I am interested in criteria that guarantee that a Borel probability measure has compact support.
I outline two below and I am hoping to gather more as answers (if they exist).
The first sufficient ...
- 850
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54
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Reference request: "doubly empirical" measure associated to a random measure
I am considering the following type of situation. Suppose we have a random probability measure, by which I mean a probability measure on a space of probability measures atop some Polish space $X$. In ...
- 660
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Concentration of measure on spheres with respect to a unitary of trace approximately zero
Cross-posted from MSE, where it hasn’t received any answer yet:
This question arose out of my attempt to understand how a unitary of trace approximately zero acts on the unit sphere of a $n$-...
- 2,830
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Directions of differentiability of log-concave measures with infinite-dimensional support
I recently came across the very nice review "Differentiable Measures and the Malliavin Calculus" by Bogachev (1997) which begins by discussing measures $\mu$ on locally convex spaces $X$ ...
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86
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From convergence of sequences to uniform convergence in probability
For $n=1, 2,\ldots$ consider a sequence of sets of ascending integers $I_n=\{\underline{i}_n,\underline{i}_n+1, \ldots, \overline{i}_n\}$, with $\underline{i}_n \to \infty$ and $\underline{i}_n=o(\...
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54
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If a probability measure is a mixture of products of its marginals, does it have finite moments?
Let $\mu$ be a Borel probability measure on $\mathbb{R}^n$. For a linear subspace $E\subset \mathbb{R}^n$, let $\mu_E$ denote the marginal of $\mu$ on $E$. The usual orthogonal complement of $E$ is ...
- 716
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35
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Deterministic multifractal measure with quadratic singular spectrum?
For a non-negative locally finite measure $\mu$ on a bounded metric space $(\Omega,\mathcal{B})$, its local Holder exponent $f(x)$ is defined as $$f(x)=\lim_{r\downarrow 0}\frac{\mu(B(x,r))}{\log r}$$
...
- 715
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124
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Density of absolutely continuous measures on a Polish Space
Consider the set of all probability measures on a Polish space $X$ (equipped with the Borel $\sigma$-field $\mathcal{B}(X)$). I am wondering if there exist conditions under which a subset of measures ...
- 185
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0
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114
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Some stability and estimate of the optimal transport map (Brenier map)
Let $\mu$ and $\nu$ be two probability measures with finite moments (in $\mathcal{P}_2(\mathbb{R})$) equipped with 2-Wasserstein distance. Let $F_\mu$, $F_\nu$ be their cumulative distribution ...
- 54
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1
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102
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Lower bounds for truncated moments of Gaussian measures on Hilbert space
Let $\mu_C$ be a centered Gaussian probability Borel measure on a real separable Hilbert space $\mathcal{H}$ with covariance operator $C$. Denote the ball with radius $r$ in $\mathcal{H}$ centered at ...
- 505
1
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1
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344
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Is the Borel-Cantelli Lemma applicable here? [duplicate]
Consider $(X_{n})_{n\in\mathbb{N}}$ a sequence of random variables taking values in the set $\mathbb{Z}_{\geq 0}$ where $\mathbb{P}(X_{n} = i) > 0 $ for every $i\in\mathbb{Z}_{\geq0}$ which are ...
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452
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Is defining measures as functionals ever insufficiently general in practice?
Crossposting from Math Stack Exchange, as it has yet to receive any answers there; the original question is here.
The way I learned measures was as set functions on a $\sigma$-algebra with certain ...
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